Wall-Cooling Effects on Hypersonic Boundary-Layer Instabilities and Transition

Linear stability analysis (LSA) and direct numerical simulations (DNS) with random inflow forcing are used to study the effects of wall cooling on boundary-layer instabilities and transition (BLT) over a sharp 7-degree half-angle cone at zero-degrees angle of attack in a Mach 6 flow representative of the Air Force Research Laboratory’s Mach-6 Ludwieg Tube. The study focuses on cones with two different isothermal wall temperatures: an uncooled warm-wall case with 𝑇𝑤 = 298 K and a cooled cold-wall case with 𝑇𝑤 = 84 K. For both wall temperatures, LSA based on the parabolized stability equations identifies Mack’s second mode as the dominant instability mechanism. The planar Mack’s second-mode waves are dominant for the cold-wall case, but the oblique Mack’s second-mode waves are more amplified than the planar ones for a large extent of the warm-wall cone. At the measured transition location of 𝑥 = 0.2 m, the most amplified waves have a disturbance azimuthal wavenumber of 𝑚 = 6, which is used to define the azimuthal domain for the DNS to compute BLT in the warm-wall case. Complementary DNS with a sufficiently large azimuthal domain confirms the amplification of the oblique waves for the warm-wall case. The DNS of the cold-wall case validates many of the predictions made by linear PSE. The second-mode frequencies are higher due to the smaller boundary layer that forms over the cooled cone. The cold-wall DNS shows a downstream shift in transition location along with diminished oblique modes and a more dominant second Mack-mode instability. The study finds that with careful tuning of the forcing amplitude, a random inflow forcing method can produce very similar spectral content to the LSA predictions and the expected second-mode dominant transition at a location similar to that of a noisy wind tunnel experiment.